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Theorem nrexdv 2500
Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)
Hypothesis
Ref Expression
nrexdv.1  |-  ( (
ph  /\  x  e.  A )  ->  -.  ps )
Assertion
Ref Expression
nrexdv  |-  ( ph  ->  -.  E. x  e.  A  ps )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem nrexdv
StepHypRef Expression
1 nrexdv.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  -.  ps )
21ralrimiva 2480 . 2  |-  ( ph  ->  A. x  e.  A  -.  ps )
3 ralnex 2401 . 2  |-  ( A. x  e.  A  -.  ps 
<->  -.  E. x  e.  A  ps )
42, 3sylib 121 1  |-  ( ph  ->  -.  E. x  e.  A  ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 1463   A.wral 2391   E.wrex 2392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-5 1406  ax-gen 1408  ax-ie2 1453  ax-4 1470  ax-17 1489
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-ral 2396  df-rex 2397
This theorem is referenced by:  ltpopr  7367  cauappcvgprlemladdru  7428  cauappcvgprlemladdrl  7429  caucvgprlemladdrl  7450  caucvgprprlemaddq  7480  dvdsle  11449
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